The New York Times Book of Mathematics: More Than 100 Years of Writing by the Numbers

By Gina Kolata and Paul Hoffman

“Some of the pieces included here are important and some are curiosities, but all are absorbing . . . Recommended for casual and serious math enthusiasts.” —Library Journal

From the archives of the world’s most famous newspaper comes a collection of its very best writing on mathematics. Big and informative, The New York Times Book of Mathematics gathers more than 110 articles written from 1892 to 2010 that cover statistics, coincidences, chaos theory, famous problems, cryptography, computers, and many other topics. Edited by Pulitzer Prize finalist and senior Times writer Gina Kolata, and featuring renowned contributors such as James Gleick, William L. Laurence, Malcolm W. Browne, George Johnson, and John Markoff, it’s a must-have for any math and science enthusiast.

“Many fascinating problems are explained in language that the layperson will understand . . . This compilation of real-world applications will interest those with an inclination toward mathematics or problem-solving.” —Publishers Weekly

• Language: English
• Publisher: Open Road Integrated Media
• Release date: Jun 4, 2013
• ISBN: 9781402793288

Paul Hoffman

Paul Hoffman is the publisher of ‘Encyclopaedia Britannica’. He is the host of the five-part PBS series ‘Great Minds of Science’ and a frequent correspondent on television shows such as ‘CBS This Morning’ and ‘The News Hour with Jim Lehrer.’ For ten years, Hoffman was the president and editor-in-chief of ‘Discover’ magazine. He is the author of ten books including ‘Archimedes’ Revenge.’ He lives in Chicago, Illinois and Woodstock, New York.

Book preview

The New York Times

BOOK OF

Mathematics

MORE THAN 100 YEARS OF

WRITING BY THE NUMBERS

An Imprint of Sterling Publishing

387 Park Avenue South

New York, NY 10016

STERLING and the distinctive Sterling logo are registered trademarks of Sterling Publishing Co., Inc.

© 2013 by The New York Times Company. All rights reserved.

All Material in this book was first published in The New York Times and is copyright © The New York Times Company. All rights reserved.

The printing, copying, redistribution, or retransmission of the Material without express written permission is prohibited.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior written permission from the publisher.

ISBN 978-1-4027-9328-8

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CONTENTS

Foreword Paul Hoffman

Introduction Gina Kolata

CHAPTER 1

What Is Mathematics?

Useful Invention or Absolute Truth: What Is Math? George Johnson

But Aren’t Truth and Beauty Supposed to be Enough? James Gleick

Mathematicians Meet Computerized Ideas Gina Kolata

Mathematicians Finally Log On James Gleick

With Major Math Proof, Brute Computers Show Flash of Reasoning Power Gina Kolata

Computers Still Can’t Do Beautiful Mathematics Gina Kolata

100 Quadrillion Calculations Later, Eureka! Gina Kolata

Theorist Applies Computer Power to Uncertainty in Statistics Gina Kolata

CHAPTER 2

Statistics, Coincidences and Surprising Facts

1-in-a-Trillion Coincidence, You Say? Not Really, Experts Find Gina Kolata

Sometimes Heavier Objects Go to the Top: Here’s Why James Gleick

Behind Monty Hall’s Doors: Puzzle, Debate and Answer? John Tierney

What If They Closed 42d Street and Nobody Noticed? Gina Kolata

Down for the Count; Why Some Numbers Are Only Very Good Guesses Gina Kolata

Could It Be? Weather Has Nothing to Do with Your Arthritis Pain? Gina Kolata

Electronics to Aid Weather Figuring Sidney Shalett

Insurance as a Study; Something of the Men Who Figure Business by Algebra

Leontief’s Contribution Leonard Silk

Many Small Events May Add Up to One Mass Extinction Malcolm W. Browne

Metric Mania John Allen Paulos

In Shuffling Cards, 7 Is a Winning Number Gina Kolata

Can Game Theory Predict When Iran Will Get the Bomb? Clive Thompson

In Modeling Risk, the Human Factor Was Left Out Steve Lohr

Playing the Odds George Johnson

Monday Puzzle: Solution to Birthday Problem Pradeep Mutalik

Just What Are Your Odds in Genetic Roulette? Go Figure Gina Kolata

The 2000 Election: The Science of Counting Gina Kolata

Prospectus; Can a Computer Program Figure Out the Market? A Former Analyst and a Mathematician Are Betting That Theirs Can Janet Stites

New Tools for the I.R.S. to Sniff Out Tax Cheats David Cay Johnston

CHAPTER 3

Famous Problems, Solved and As Yet Unsolved

New Mathematics Links Two Worlds William L. Laurence

An Elusive Proof and Its Elusive Prover Dennis Overbye

Ask Science: Poincaré’s Conjecture Dennis Overbye

Grigori Perelman’s Beautiful Mind Jascha Hoffman

A Math Problem Solver Declines a $1 Million Prize Dennis Overbye

Four-Color Problem Attacked William L. Laurence

Four-Color Proof

Goldbach’s Conjecture; This One May Be Provable, but We May Never Know George Johnson

Mathematics Expert May Soon Resolve A 350-Year Problem James Gleick

Fermat’s Theorem Solved? Not This Time James Gleick

Fermat’s Last Theorem Still Has 0 Solutions James Gleick

At Last, Shout of Eureka! in Age-Old Math Mystery Gina Kolata

Fermat’s Theorem James Gleick

Flaw Is Found in Math Proof, but Repairs Are Under Way Gina Kolata

A Year Later Fermat’s Puzzle Is Still Not Quite Q.E.D. Gina Kolata

How a Gap in the Fermat Proof Was Bridged Gina Kolata

Two Key Mathematics Questions Answered after Quarter Century John A. Osmundsen

Mathematical Theory of Poker Is Applied to Business Problems Will Lissner

Soap Bubbles Get a New Role in Old Mathematics Problem Joseph Williams

Math Advance Penetrates Secrets of Knots James Gleick

Packing Tetrahedrons, and Closing in on a Perfect Fit Kenneth Chang

Finding Order in the Apparent Chaos of Currents Bina Venkataraman

In Bubbles and Metal, the Art of Shape-Shifting Kenneth Chang

The Scientific Promise of Perfect Symmetry Kenneth Chang

143-Year-Old Problem Still Has Mathematicians Guessing Bruce Schechter

What Is the Most Important Problem in Math Today? Gina Kolata

Solution to Old Puzzle: How Short a Shortcut? Gina Kolata

CHAPTER 4

Chaos, Catastrophe and Randomness

Chaos Is Defined by New Calculus

Experts Debate the Prediction of Disasters Malcolm W. Browne

Solving the Mathematical Riddle of Chaos James Gleick

The Man Who Reshaped Geometry James Gleick

Snowflake’s Riddle Yields to Probing of Science James Gleick

Tales of Chaos: Tumbling Moons and Unstable Asteroids James Gleick

Fluid Math Made Simple—Sort Of James Gleick

When Chaos Rules the Market James Gleick

New Appreciation of the Complexity in a Flock of Birds James Gleick

Indestructible Wave May Hold Key to Superconductors James Gleick

The Quest for True Randomness Finally Appears Successful James Gleick

Coin-Tossing Computers Found to Show Subtle Bias Malcolm W. Browne

Science Squints at a Future Fogged by Chaotic Uncertainty Malcolm W. Browne

Probing Disease Clusters: Easier to Spot Than Prove Gina Kolata

The Odds of That Lisa Belkin

Fractal Vision James Gleick

CHAPTER 5

Cryptography and the Emergence of Truly Unbreakable Codes

Harassment Alleged over Code Research Malcolm W. Browne

Researchers to Permit Pre-Publication Review by U.S. Richard Severo

Tighter Security Rules for Advances in Cryptology Walter Sullivan

A New Approach to Protecting Secrets Is Discovered James Gleick

Brief U.S. Suppression of Proof Stirs Anger

A Most Ferocious Math Problem Tamed Malcolm W. Browne

Biggest Division a Giant Leap in Math Gina Kolata

Scientists Devise Math Tool to Break a Protective Code John Markoff

Tied Up in Knots, Cryptographers Test Their Limits Gina Kolata

A Public Battle over Secret Codes John Markoff

U.S. Code Agency Is Jostling for Civilian Turf John Markoff

Researchers Demonstrate Computer Code Can Be Broken Sara Robinson

Nick Patterson; A Cold War Cryptologist Takes a Crack at Deciphering DNA’s Deep Secrets Ingfei Chen

Adding Math to List of Security Threats John Markoff

Prizes Aside, the P-NP Puzzler Has Consequences John Markoff

CHAPTER 6

Computers Enter the World of Mathematics

Thinking Machine Does Higher Mathematics; Solves Equations That Take Humans Months

New Giant Brain Does Wizard Work

Brain Speeded Up for War Problems Will Lissner

The Electronic Digital Computer: How It Started, How It Works and What It Does Henry L. Lieberman and Dr. Louis Robinson

New Shortcut Found for Long Math Proofs Gina Kolata

New Technique Stores Images More Efficiently Gina Kolata

Giant Computer Virtually Conquers Space and Time George Johnson

Rear Adm. Grace M. Hopper Dies; Innovator in Computers Was 85 John Markoff

Frances E. Holberton, 84, Early Computer Programmer Steve Lohr

Squeezing Data like an Accordion Peter Wayner

A Digital Brain Makes Connections Anne Eisenberg

A Soviet Discovery Rocks World of Mathematics Malcolm W. Browne

The Health Care Debate: Finding What Works Gina Kolata

Step 1: Post Elusive Proof. Step 2: Watch Fireworks. John Markoff

CHAPTER 7

Mathematicians and Their World

Paul Erdos, 83, a Wayfarer in Math’s Vanguard, Is Dead Gina Kolata

Journeys to the Distant Fields of Prime Kenneth Chang

Highest Honor in Mathematics Is Refused Kenneth Chang

Scientist at Work: John H. Conway; At Home in the Elusive World of Mathematics Gina Kolata

Claude Shannon, B. 1916—Bit Player James Gleick

An Isolated Genius Is Given His Due James Gleick

Scientist at Work: Andrew Wiles; Math Whiz Who Battled 350-Year-Old Problem Gina Kolata

Scientist at Work: Leonard Adleman; Hitting the High Spots of Computer Theory Gina Kolata

Dr. Kurt Gödel, 71, Mathematician Peter B. Flint

Genius or Gibberish? The Strange World of the Math Crank George Johnson

Contributors’ Biographies

Photography and Illustration Credits

Ackowledgments

Index

FOREWORD

A formidable student at Trinity

Solved the square root of infinity;

It gave him such fidgets

To count up the digits

He chucked math and took up divinity.

—Anonymous

This old limerick, penned by an anonymous wag, encapsulates two fundamental characteristics of mathematics and at least one misconception. The limerick correctly portrays mathematics as addressing big, head-spinning notions like that of infinity, and it gets across the fact that mathematical concepts can be difficult for laymen to comprehend. But the verse wrongly suggests that counting is the essence of mathematics. Counting is for bank tellers, cashiers, and CPAs. As the Times articles in this book make clear, mathematicians do much more than arithmetic. They are on the hunt for eternal universal truths, a quest that a budding theologian at Trinity would appreciate.

Around 300 B.C., Euclid of Alexandria offered a simple proof that there are infinitely many prime numbers (integers like 3, 5, 7, 11, and 13 that are divisible only by themselves and the number 1). His proof is as true today as it was twenty-three hundred years ago, and it will be true twenty-three hundred years from now. This kind of categorical certainty makes mathematics unique among the sciences. No matter how far out we go in the number line (beyond a hundred million, or a hundred million million, or a hundred million million million) there will always be primes. The truths of physics are not so universal. The discoveries of Newton and Einstein seem to hold only at certain familiar scales and not at far out distances.

More than the other sciences, mathematics seems to pose an abundance of simple questions that are beyond current human ingenuity. To take but one example in number theory, there is the unsolved puzzle of whether the supply of so-called twin primes (pairs of primes that differ by two, such as 3 and 5, 17 and 19, and 3,581 and 3,583) is inexhaustible. Although Euclid proved that the number of primes is infinite, generations of mathematicians have not been able to prove their suspicion that the number of twin primes is also infinite. Euclid’s proof is so simple that it can be written on one side of an index card. This raises the deep meta-mathematical question of why certain seemingly simple mathematical questions are maddeningly intractable compared to other related questions that readily give up their answers.

Contemporary mathematics is still about the search for eternal truths, but mathematicians now have a wider repertoire of techniques for getting at these truths. Chief among these is the computer, which itself is the product of mathematics, the binary language of 0s and 1s. I was in college in the mid 1970s when my professors at Harvard received word that two mathematicians in the Midwest had proved the century-old four-color conjecture, which asserts that any imaginary map of countries can always be colored with at most four colors in such a way that no two bordering countries share the same color. A lot of champagne was poured in the Harvard math department, but jubilation later gave way to consternation when word got out that the proof involved a computer doing much of the heavy lifting. It wasn’t that my professors distrusted the proof; they presumed that the correctness of its numerous steps could be confirmed by running the proof on another machine. They were disquieted by the fact that the proof was so long that no human being would be able to assimilate it and understand why four colors were sufficient. For them a proof had to provide insight, and in this regard Euclid’s half-page proof was the gold standard: they could easily follow it and understand why the number of primes was infinite.

You’ll learn from the Times articles within about the four-color conjecture and other intriguing mathematical conundrums that computers have helped to solve (for example, the party planner’s problem of determining the minimum number of guests that must be invited to guarantee that eight or more guests all know one another and that three or more will be strangers) and the meta issues such computer-aided proofs raise. You’ll also find stories about famous problems such as Fermat’s Last Theorem and the Poincaré conjecture that were recently proved the old-fashioned way, without depending on machines.

The men you’ll meet in these pages (and I say men, because few of these mathematicians are women) include other worldly eccentrics like Grigori Perelman, who solved the century-old Poincaré conjecture only to retreat into the Russian woods, turning down both a $1-million reward and the Fields Medal, the equivalent in mathematics of the Nobel Prize, and Paul Erdos, arguably the world’s most prolific mathematician, a homeless Hungarian who, jacked up on amphetamines and espresso, worked on problems twenty hours a day and published more than 1,500 papers. Mathematicians are machines for turning coffee into theorems, Erdos would say. When colleagues asked him to slow down, he’d reply, They’ll be plenty of time to rest in the grave. Erdos died at the age of eighty-three, but his mathematics, like all mathematics, is immortal. The discoveries he made, like Euclid’s and Grigori’s, will hold up for eternity.

Paul Hoffman

CEO of Liberty Science Center and author of The Man Who Loved Only Numbers

INTRODUCTION

Amathematician once dismissed the very idea that people outside his circle could ever understand the true essence of the field. Mathematics is an art form, like music or painting. Translating math into the English language, he said, is harder than translating Chinese poetry. The beauty is lost, the elegance, and a proof that is a thing of ineffable iridescence becomes reduced to a baffling or mundane-sounding bottom line.

Others echoed that sentiment. Fritz John, a mathematician at New York University, said he wanted neither fame nor fortune but merely the grudging admiration of a few close friends.

But even if the rest of us cannot appreciate mathematics as an art form, are we really shut out? Articles in The New York Times may not give the details of proofs, but they reveal a rich world that can be exciting, surprising, and can even tug at the heartstrings. They even address the age-old question, What is mathematics? Is it discovered or is it invented? Art or science? If it is art, then why, as George Johnson wrote in one of the articles that opens this book, does the universe appear to follow mathematical laws?

Yet if we put philosophy aside, the variety of mathematical questions, the scope of its inquiries as reported in the Times, can be stunning. There are articles that give mathematical solutions to everyday questions—why do heavy objects rise to the top of a container? The strawberries in your jam are all at the top of the jar so when you get toward the bottom, all you have is thick syrup. Brazil nuts are at the top of the mixed nuts can. A mathematical discovery tells why.

Or what about the woman who won the New Jersey lottery twice in four months? The odds of that happening were widely reported as 1 in 17 trillion. But actually, statisticians calculated, they were more like 1 in 30. A Times article tells how to reason through such questions.

And does arthritis pain really respond to changes in the weather? Statisticians answer that one, too, with an analysis that makes sense but confounds perceptions. Another article tells of a surprising result in controlling traffic jams—mathematicians can prove that sometimes closing streets actually improves traffic flow.

But while those articles can make us look at the world differently, they are not about results that rocked mathematics. If you asked mathematicians which proofs were most important to them, many would cite the surprising, drama-laden tales of the search for solutions to some of their most famous problems. And the story of Fermat’s Last Theorem would be at the top of many lists. Nearly 400 years ago, a French mathematician scrawled the problem in the margins of a book, saying he had a simple proof but no space to write it. Ever since, mathematicians tried to solve it, to no avail. Some famous mathematicians said they would not even take it up—it was a fool’s errand, they would just waste precious years of their lives only to end up empty-handed.

Then, in 1993, a young mathematician, Andrew Wiles, announced that he had solved Fermat. What followed was elation, followed by intense questioning. It was a complicated proof, relying on recently discovered mathematics that few truly understood. And as mathematicians scrutinized Wiles’s work, they found a hole in the proof, which Wiles then desperately tried to fix. He retreated to a barren office in his attic, where he had secretly done his work, attempting to make the proof whole again. It was a year of drama that ended well, but the rollercoaster tale, told in the pages of the Times as it happened, was an unforgettable story of pride and ambition, talent and determination.

Not all mathematics is logical, and researchers have wondered how to describe the unpredictable, like the famous analogy of the Butterfly Effect, described by mathematician Edward Lorenz in a lecture in 1972 titled Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas? As mathematicians tried to understand such phenomena, they developed the new fields, or newly named fields, chaos theory and catastrophe theory. And with those fields came debate. Were the results overstated? How predictable were some of these events? And what can the research say about important questions, like global warming, which involve some of the same uncertainties?

Other articles involve discoveries that changed the way we live and work. In the 1980s, a few mathematicians had a brilliant idea for making unbreakable secret codes. There are some problems that are simple to check once you have the answer, but pretty much impossible to solve unless you have centuries or millennia to run your computer. One such problem is factoring—figuring out which prime numbers can be multiplied together to make a particular number. You can easily multiply, say 3 times 5 times 11 times 13 to get 2,145, but it is a bit more work to start with 2,145 and find those factors. The codes, though, use enormous numbers, not simple ones like 2,145. And with huge numbers, there is no easy way to get the factors. So, suppose you made a code that required you to factor a large number if you wanted to break it. You could send a message by doing the equivalent of multiplying a large group of prime numbers together. And no one could illicitly read that message without factoring the resulting huge number.

The idea was so powerful that the federal government got alarmed at mathematicians’ proclivities to publish all of their work, leading to a difficult national debate about how much can or should be revealed. Some said it was important for the codes to be public in order to use them to keep sensitive information, like credit card transactions, private. Others said it was important to keep coding methods secret so enemies could not use them and make codes that the government could not break. In the end, the methods became an integral part of today’s online world, allowing, for example, the secure Web sites we use when we shop online. The Times articles tell the story of the discoveries and the wrenching debate.

Although many fields of science today involve huge teams of researchers, mathematicians often work alone. One brilliant person can change a field. And many of these mathematicians have stories and insights that can be unforgettable. The Times articles include the haunting story of Srinivasa Ramanujan, born in the 19th century in a small town in India, who died at age 32. He left a strange, raw legacy, about 4,000 formulas written on the pages of three notebooks and some scrap paper. His extraordinary story—how he was discovered, how inventive he was—is a tale like no other. He seemed to have functioned in a way unlike anybody else we know of, one mathematician said. He had such a feel for things that they just flowed out of his brain. Perhaps he didn’t see them in way that’s translatable. It’s like watching somebody at a feast that you haven’t been invited to.

Contemporary mathematicians can tell us what it might feel like to be part of their feast. Leonard Adleman, one of the inventors of the new type of secret code, discovered in graduate school that mathematics is less related to accounting than it is to philosophy. While many think of mathematics as some kind of practical art, he said, the point when you become a mathematician is when you somehow see through this and see the beauty and power of mathematics.

Some, not surprisingly, are just odd people, geniuses but eccentric almost beyond belief. That, at least, describes Paul Erdos, a Hungarian mathematician who had no home and no job. Other mathematicians invited him into their homes, feeding and housing him—and collaborating with him—until he moved on to another mathematician’s home. He also took on the question that opens this book. Are mathematical truths discovered or invented? Erdos said they were discovered. As I wrote in an obituary about this unforgettable man, Erdos spoke of a Great Book in the sky, maintained by God, that contained the most elegant proofs of every mathematical problem. He used to joke about what he might find if he could just have a glimpse of that book.

Gina Kolata

CHAPTER 1

What Is Mathematics?

MATHEMATICIANS SOMETIMES ASK THEMSELVES IF THEY are finding truths about the real world or indulging in creative inventions. Is mathematics an art or a science? Often someone will cite an article by the physicist Eugene Wigner, published in 1960 and titled The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics, Wigner wrote, is almost uncannily effective in describing physical phenomena. And so, he wrote, The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

In this section, George Johnson takes this issue of the nature of mathematics. He tells of the argument of an eminent mathematician, Reuben Hersh, that mathematics is invented. He quotes Einstein, who said humans invented mathematical tools that allowed us to describe nature but were not, in fact, an essential truth.

Jim Gleick addresses a closely related issue. What is mathematics good for? Is it enough to say that mathematics, discovered or invented, should just be pursued for its own sake, with applications coming later? Is it enough to evoke the Wigner argument—if mathematical truths are found they will turn out to be unreasonably effective in describing the natural world?

Or what if a computer, an unthinking computer, could spew out mathematical ideas or even produce proofs that would be considered creative if a human produced them? What does that do to the idea of mathematics as an art form?

There is no consensus in these articles, but they abound with provocative insights and questions.

Useful Invention or Absolute Truth: What Is Math?

By GEORGE JOHNSON

At the top of the list of science’s unanswered questions, like what is consciousness and how did life begin, is the deepest mystery of all: Why does the universe appear to follow mathematical laws?

According to the Big Bang theory, matter, energy, space and time were created during the primeval explosion. Instantly, it seems, everything began unfolding according to a mathematical plan. But where did the mathematics come from? What are the origins of numbers and the relationships they obey?

The ancient followers of the Greek mathematician Pythagoras declared that numbers were the basic elements of the universe. Ever since, scientists have embraced a kind of mathematical creationism: God is a great mathematician, who declared, Let there be numbers! before getting around to let there be light!

Scientists usually use the notion of God metaphorically. But ultimately, most of them at least tacitly embrace the philosophy of Plato, who proposed, rather unscientifically, that numbers and mathematical laws are ethereal ideals, existing outside of space and time in a realm beyond the reach of humankind.

Because the whole point of science is to describe the universe without invoking the supernatural, the failure to explain rationally the unreasonable effectiveness of mathematics, as the physicist Eugene Wigner once put it, is something of a scandal, an enormous gap in human understanding.

We refuse to face this embarrassment, Reuben Hersh, a mathematician emeritus of the University of New Mexico in Albuquerque, wrote in his recent book, What Is Mathematics Really? (Oxford University Press, 1997). Ideal entities independent of human consciousness violate the empiricism of modern science. While science is anchored in observations of the physical world, Dr. Hersh insists that mathematics is more of a human creation, like literature, religion or banking.

Dr. Hersh’s book is one of several recent works contending that mathematics is not an ethereal essence but comes from people who invented, not discovered, it. The sentiments presented in the books are not entirely new and the mathematical puzzle has hardly been solved. But the idea of a human-centered mathematics may be gaining force and respect.

In The Number Sense: How the Mind Creates Mathematics (Oxford University Press, 1997), Stanislas Dehaene, a cognitive scientist at the National Institute of Health and Medical Research in Paris, marshals experimental evidence to show that the brains of humans—and even of chimpanzees and rats—may come equipped at birth with an innate, wired-in aptitude for mathematics. Gregory J. Chaitin, a mathematician at I.B.M.’s Thomas J. Watson Research Center in Yorktown Heights, N.Y., takes an anti-Platonist stance in The Limits of Mathematics (Springer, 1997). Two Berkeley scientists, George Lakoff and Rafael E. Núñez, are working on a book tentatively called The Mathematical Body, contending that even the most abstract mathematical concepts arise from basic human experience—from the way the body interacts with the world. They gave a preview of their ideas in a chapter of another book published last year: Mathematical Reasoning: Analogies, Metaphors and Images, edited by Lyn D. English (Erlbaum).

The authors are all working mathematicians and scientists, not postmodern critics viewing the territory from afar. They emphatically reject those who try to dismiss mathematics and science as arbitrary constructions, or white male Eurocentric folklore. But they are just as adamant in rejecting what most mathematicians and many scientists have come to take for granted: the Platonic creed.

The normal notion of pure math is that mathematicians have some kind of direct pipeline to God’s thoughts, to absolute truth, Dr. Chaitin wrote in The Limits of Mathematics. While scientific knowledge is tentative and subject to constant revision, mathematics is usually seen as eternal. But Dr. Chaitin called on his colleagues to abandon mathematical Platonism and adopt a quasi-empirical approach that treats mathematics as just another messy experimental science.

Quasi-empirical, he said, means that math ain’t that different from physics. This view is laid out in detail in a revised edition of New Directions in the Philosophy of Mathematics, edited by Thomas Tymoczco (Princeton University Press, 1998).

Leopold Kronecker, a 19th-century mathematician, once said: The integers were created by God; all else is the work of man. Albert Einstein, taking a different view of whole numbers, wrote that the series of integers is obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences.

In The Number Sense, Dr. Dehaene went even further. The integers—the smallest ones, anyway—are hard-wired into human nervous systems by evolution, along with a crude ability to add and subtract. Mathematics, he believes, is engraved in the very architecture of our brains.

Because we live in a world full of discrete and movable objects, it is very useful for us to be able to extract number, he argued in a recent forum published on the Internet (www.edge.org) by the Edge Foundation. This can help us to track predators or to select the best foraging grounds, to mention only very obvious examples.

By studying brain-damaged patients who have lost basic number skills, Dr. Dehaene and others have tentatively traced this arithmetical module to an area of the brain called the inferior parietal cortex, a poorly understood location where visual, auditory and tactile signals converge. Scientists are intrigued by clues that this region is also involved in language processing and in distinguishing right from left. Mathematics is, after all, a kind of language intimately involved with using numbers to order space. The inferior parietal cortex also seems to be important for manual dexterity, and arithmetic begins with counting on the hands. Imaging experiments, in which people’s brains are monitored as they calculate, point to the same region as a primitive number processor.

If this neurological calculator has indeed been bequeathed by evolution, then traces of it should be found in other species. In making his argument, Dr. Dehaene draws on experiments over the last few decades suggesting that even rats have a rudimentary number sense. The animals were taught to press lever A four times and then lever B to get food, or to press lever A when they heard a two-tone sequence and lever B when they heard an eight-tone sequence. (To insure that the rats were responding to the number of signals and not just to their duration, the two-tone sequence sometimes lasted longer than the eight-tone one.)

Even more striking were later experiments in which rats were first trained to associate lever A with two tones and lever B with four tones. Then they were taught to associate A with two flashes of light and B with four flashes. If the rats heard two tones and saw two flashes they learned to push B, not A. They seemed to have comprehended the notion that two plus two equals four.

The rats were not precise. Trained to press one lever four times, they often pressed it five or six times, expecting to be rewarded just the same, or they confused a seven-tone sequence with an eight-tone one. But the experiments support the notion of a primitive neurological number processor, even in rodents.

In other experiments, chimpanzees seemed to learn simple arithmetic. Given a choice between one tray with a pile of three chocolate chips and another pile of four and a second tray with piles of two and three chips, they chose the first tray with the most candy. But when the totals on the trays differed by only one chip, the chimps were less likely to make the discrimination. The number sense is approximate, not exact. More recent experiments on infants, using Mickey Mouse toys instead of chocolate chips, found signs of the same kind of rough numerical ability in babies less than 5 months old.

Dr. Dehaene says this instinct is innate, as singing is for songbirds or spinning webs is for spiders. Numbers are not Platonic ideals but neurological creations, artifacts of the way the brain parses the world. In that sense they are like colors. Red apples are not inherently red. They reflect light at wavelengths that the brain, as it was wired by evolution, interprets as red.

While people are born with an understanding of the rudiments of arithmetic, he contends, going beyond that requires learning and creativity. Multiplication, division and the whole superstructure of higher mathematics—from algebra and trigonometry, to calculus, fractal geometry and beyond—are a beautiful improvisation, the work of human culture.

The ability to weave simple ideas, like two plus two equals four, into the tapestries of higher mathematics, he suggests, is not unlike the human skill for language. People take a relatively small collection of words and, using a few simple rules of grammar and syntax, create literature.

At the University of California at Berkeley, Dr. Lakoff, a linguist and cognitive scientist, and Dr. Núñez, a developmental psychologist, contend that the source of mathematics lies not just in the brain but in the human body and the physical world. People favor number systems based on 10 because they have 10 fingers and 10 toes. But that is just the beginning of the story.

Driven by a built-in number sense, the theory goes, primitive people explored the wonders of counting by playing with their fingers or putting rocks in a pile. But they found that counting could also be thought of as taking steps along a line to measure distance. That metaphor eventually allowed for the invention of more abstract concepts. Walk one way and you get the positive integers; walk the other way and you get the negative integers. The starting point is zero.

Multiplication by a positive number can be thought of as stretching; multiplying by a negative number makes something shrink.

Dr. Lakoff and Dr. Núñez call these grounding metaphors. In inventing mathematics, they contend, people also used linking metaphors to connect two sets of ideas. The sequence of numbers can be mapped onto the notion of a line. Now numbers are not fingers or rocks but points. Put two lines together at right angles and you get what mathematicians call a Cartesian plane, a twodimensional graph that opens up a whole new arena to play in.

And so, floor by floor, the tower of mathematics is built. Students never learn that mathematics is a creative endeavor, Dr. Lakoff said in a recent interview. Mathematics is more glorious because it is humanly constructed. There is no such thing as pure mathematics or pure thought, he said—they are physical activities.

That does not mean that mathematics is a relativistic free-for-all. The most basic mathematical inventions are rooted in the brain and body. Even mathematicians’ loftier elaborations are tested against the universe. Of the infinite range of mathematical creations, scientists keep those that help them explain and predict reality. Mathematicians savor the others as ends in themselves, like paintings or symphonies.

But many scientists and mathematicians still doubt that evolution—biological or cultural—can adequately explain why mathematics works so well in describing the fundamental laws of the universe.

Our ability to discover, and describe mathematically, Newton’s equations has no immediate survival value, said Dr. Paul Davies, professor of mathematical physics at the University of Adelaide in Australia. This point has even greater force when it comes to, say, quantum mechanics. The reason people find it hard to understand quantum physics is precisely because there is no survival value in being able to do so.

The reason mathematics is so effective, he says, remains a deep mystery. No feature of this uncanny ‘tuning’ of the human mind to the workings of nature is more striking than mathematics, he wrote in The Mind of God: The Scientific Basis for a Rational World (Simon & Schuster, 1992).

Some hold out vague hopes that the mystery might be solved if humans ever encounter an alien civilization. If mathematics is indeed universal and eternal, the theory goes, then the aliens would understand concepts like pi, the ratio of a circle’s circumference to its diameter. The Platonists assume that there is pi in the sky, as the British astronomer John D. Barrow said in a book by that name (Oxford University Press, 1992).

The anti-Platonists say there is no reason to believe the aliens would understand mathematical inventions from Earth. The Platonist claim that every intelligence must produce prime numbers, pi and the continuum hypothesis is an example of simple anthropomorphism, Dr. Hersh said.

But if earthlings were utterly baffled by extraterrestrial mathematics, would the anti-Platonists have proved their point? Not necessarily.

Alien intelligences may be so far advanced that their math would simply be too hard for us to grasp, Dr. Davies said. The calculus would have baffled Pythagoras, but with suitable tuition he would have accepted it.

But what if the humans and the aliens could communicate mathematically? Would that decide the issue in favor of the Platonists? Not really.

If the alien species had evolved in an environment similar to ours—say, a world composed of distinct, movable objects—then most likely its brain would have incorporated, through natural selection, the same regularities about the external world as we have, Dr. Dehaene said. "Thus, it would have a very similar arithmetic and geometry.

But now, suppose that the alien species has evolved in a radically different environment, like a fluid world, he continued. Then knowledge of movable objects would not be essential to its survival, while knowledge of fluid mechanics, vortices, etc. would be. I believe that this hypothetical species would have internalized in its brain regularities strikingly different from ours. Hence it would have radically different mathematics.

And so the argument continues to churn.

Several years ago, the French mathematician Alain Connes, arguing for the Platonists, and the French neurobiologist Jean-Pierre Changeux, taking the opposite side, tried to settle the matter with a debate. The result, translated and edited by M. B. DeBevoise, was the book Conversations on Mind, Matter and Mathematics (Princeton University Press, 1995).

Ranging over a vast field of topics including relativity, quantum mechanics, neurobiology, topology, game theory, information theory and non-Euclidean geometry, the two reached the end of their discussion with no resolution.

The best they could do was to agree to disagree.

February 11, 1998

But Aren’t Truth and Beauty Supposed to be Enough?

By JAMES GLEICK

Could the mathematicians, winners of the most prestigious awards of their discipline, please tell the audience what their work is good for?

Flush with pleasure, these four young men, carrying home three Fields Medals and a Nevanlinna Prize, were telling a lay audience what their work was about. Two had discovered astonishing facts about shapes in four-dimensional space. One had developed important insights into what makes hard problems hard. One had proved the Mordell conjecture, the idea that a large class of equations can have only a finite number of rational solutions.

To the nearly 4,000 mathematicians who gathered here for the International Congress of Mathematicians, which ended Monday, these were an astonishing set of breakthroughs demonstrating new vitality in the purest of sciences. But a reporter-cameraman for a local television station wanted at least one of the prize winners to address a basic question: How would their achievements improve life for the viewers at home?

Embarrassed silence. The mathematicians suddenly seemed to have remembered pressing engagements elsewhere. They looked at one another. Gerd Faltings, a boyish, blond West German who became one of mathematics’ great men when he proved the Mordell conjecture, gave an awkward smile and flatly refused to speak.

It was Michael Freedman, a topologist in California, who rose in the end to say what all the mathematicians felt: That theirs is a way of thinking that thrives by disdaining the need for practical applications. Let the applications come later by accident—they always do. A weird, curved parody of Euclidean geometry turns out to be just the framework a physicist needs to invent the General Theory of Relativity. Notoriously unpractical techniques of number theory turn out to be just what the National Security Agency needs to make efficient, secure codes.

Usually unspoken, but always present, is the faith that doing mathematics purely, following an internal compass, seeking elegance and beauty in a strange abstract world, is the best way in the long run to serve practical science. As physics or biology progress, they will inevitably find that the way ahead has been cleared by some odd piece of pure mathematics that was thought dead and buried for many decades.

We’re a part of a gigantic enterprise that has gone on for hundreds of years, and interacts in interesting ways with science, and operates on a very low budget, Dr. Freedman said, and we’ve learned that it’s hard to prophesy what piece of mathematics will have what particular applications. Mathematics has to advance as an organic whole, in ways that seem right to the people inside it.

An older mathematician, Sir Michael Atiyah of Oxford University, who won a Fields Medal himself in 1966, offered one correction. It’s been thousands of years, he said. So we’re in business on a long-term basis.

Yet the meaning of mathematical purity is changing—has changed, many mathematicians said, even since the last Congress in Warsaw in 1983. Questions about the nature of that change, and what it might mean for the future, hovered in the air through 16 plenary addresses, scores of 45-minute lectures and hundreds of 10-minute short communications in the nine-day conference.

Mostly unrepresented was the somewhat less exalted discipline known as applied mathematics, the traditional route for mathematical ideas to filter down to engineering and other sciences. A few mathematicians could not help noticing, though, that recently physical scientists have been plucking ideas directly from the heart of pure mathematics, bypassing applied mathematics altogether. Many unexpected connections have arisen—between knot theory and genetic processes in DNA strands, for example—but the most important has been the use of geometric ideas in the theory of cosmic strings, the hottest new game in the physics of fundamental forces and particles.

The suspicion of a few mathematicians here was that biologists and chemists can no longer be relied on to be naive about the arcana of number theory or topology. That will take some getting used to.

And in the case of strings, the physics has begun feeding back into mathematics, meaning that the up-to-date pure mathematician may now have to learn some unpure science. This state of affairs was highlighted by two unusual talks by physicists Edward Witten of Princeton University and Aleksandr Polyakov of the Soviet Union’s Landau Institute for Theoretical Physics.

Dr. Polyakov, an intense man with long sandy hair, paced nervously before his lecture, a red knapsack on his back. He was worried that his mathematician audience would be put off by having to hear a foreign language—physics, not Russian.

I apologize if you are irritated by the reckless manner of a physicist, he told his audience. Reckless, because the two disciplines have different standards of proof: A physicist is content to say that the earth orbits the sun; a mathematician will say only that there is convincing evidence.

Dr. Freedman, in the work that won his Fields Medal, proved that certain exotic four-dimensional spaces exist. Another medalist, Simon Donaldson of Oxford, meanwhile, used tools from physics to prove that these same spaces could not exist.

So the conclusion a mathematician would draw, Dr. Freedman said, is that physics doesn’t exist.

To some mathematicians, purity has always meant a certain degree of inscrutability. That, at least, has not changed.

Sometimes inscrutability comes with the territory—for example, when the territory has four dimensions or more. A mathematician needs to be comfortable with shapes in many dimensions, but not everyone can actually visualize more than the usual three. That is one reason geometry relies, for the sake of purity, on rigorous proofs using numbers and symbols. Visual imagination cannot be trusted.

One dimension is a line. The second dimension comes when you add a second line at right angles to the first, so that now you have east-west and north-south. The third dimension requires a new line at right angles to the others, so you must leave the flat plain and draw one up-down. To imagine a fourth dimension, it is necessary to imagine a fourth line at right angles to all the others, and this most mortals cannot do.

Yet some kind of inner vision led John Milnor of the Institute for Advanced Study in Princeton, in describing the four-dimensional discoveries of Dr. Freedman to an audience of several thousand, to start gesturing with his hands.

The problem is, he was saying, when you try to embed a two-dimensional disk inside a four-dimensional manifold, it will usually intersect itself. His hands formed loops and handles in the air, as though he were describing some new kind of suitcase.

Sometimes inscrutability is just a matter of style.

It has been said that the ideal mathematics talk has three parts. The first part should be understood by most of your audience. The second part should be understood by four or five specialists in your field. The third part should be understood by no one—because how else will people know you are serious?

Some speakers seemed to follow these guidelines, mathematicians felt. Others, perhaps to save time, skipped directly to part three.

There was one question that the Fields Medalists could not wait to answer, and that was whether they used computers, the unloved child of mathematics and an object whose influence was more in evidence at the Congress than ever before.

No, Dr. Freedman said. Dr. Donaldson: No. Dr. Faltings said, Perhaps it could reduce some sorts of tedious work for us, but it doesn’t do the thinking. Personally he doesn’t use one.

Eyes turned to Leslie G. Valiant of Harvard University, winner of the recently established Nevanlinna Prize for Information Science, whose work centered on computer algorithms. Maybe I should clarify my own position, Dr. Valiant said. I don’t use computers either.

If the mathematicians were inclined toward parable and metaphor—and they most definitely are not—they might describe a vast wilderness, and in it a small society of men and women whose business it is to lay railroad track. This has become an art, and they have become artists—artists of track, lovers of track, connoisseurs of track.

Almost perversely, they ignore the landscape around them. A network of track may head to the northeast for many years and then be abandoned. An old, nearly forgotten line to the south may sprout new branches, heading toward a horizon that the tracklayers seem unable or unwilling to see.

As long as each new piece of track is carefully joined to the old, so that the progression is never broken, an odd thing happens. People come along hoping to explore this forest or that desert, and they find that a certain stretch of track takes them exactly where they need to go. The tracklayers, for their part, may have long since abandoned that place. But the track remains, and track, of course, is the stuff on which the engines of knowledge roll forward.

August 12, 1986

Mathematicians Meet Computerized Ideas

By GINA KOLATA

Suppose a computer program could give artists their ideas for what to paint.

Or what if a computer program churned out possible story ideas for novelists? Would anyone want to use them?

Mathematicians, many of whom consider themselves more artists than scientists, have had to consider exactly this problem. A researcher has devised a program that spews out conjectures in a field of mathematics known as graph theory.

A conjecture is to a mathematician what a hypothesis is to a scientist: an educated guess to be tested for possible canonization into the realm of truth.

Finding a good conjecture is definitely half of the job, said Siemion Fajtlowicz, the University of Houston mathematics professor who wrote the program, called Graffiti. The program can easily generate tens of thousands of conjectures, he said. Although some mathematicians have been happy to work on proving the conjectures true or false, others seem to resent the computer, Dr. Fajtlowicz said. I’ve run into quite a few Luddites, he said.

Although Dr. Fajtlowicz has been promoting his program at math meetings, throwing out conjectures and hoping that his colleagues or students would become interested in trying to prove them, most mathematicians, even graph theorists, have still not heard of it. It remains a somewhat bizarre source of mathematical inspiration and an irritant to some researchers.

One mathematician told Dr. Fajtlowicz he was furious that the computer had chanced upon a mathematical truth that he himself had discovered. Another mathematician wrote to Dr. Fajtlowicz, only half in jest, that Graffiti was putting him out of a job.

Seeking Truths

Dr. Fajtlowicz said his computer program starts with a collection of graphs, which are groups of points connected by lines. A highway road map is a graph, for example. The program looks for relationships that seem to hold true for the graphs in its collection and then proposes them as more general truths.

In a way, it is like the story of the monkeys banging away at typewriters. Sooner or later, one will type Hamlet. And sooner or later, Graffiti will find important mathematical truths by randomly trying many different relationships. But the challenge, Dr. Fajtlowicz said, is to find that mathematical work of art when it appears. "The difficult thing is to discard everything else but Hamlet," he said.

So Dr. Fajtlowicz had to decide what made a good conjecture. He said he could not simply ask the program to spit out every conjecture it could find because he would soon be buried in a pile of mostly uninteresting ones. One round of Graffiti results in 3,000 to 8,000 conjectures.

The trick was to find a way to automate a mathematician’s instinctive feeling that some possibilities are more worthy of study than others. But mathematicians describe the act of research as highly creative and say they are guided by an intuitive sense of what is significant and what is not. They often describe a theorem they like as beautiful and a particularly striking proof as elegant.

I asked everybody in sight, ‘How do you know if a conjecture is interesting?’ Dr. Fajtlowicz said. Nobody seemed to know.

He eventually chose four criteria. A conjecture would have to be surprising, judged by how different it was from conjectures the computer had made before. It could not be a logical consequence of another conjecture. It should not be overly specific. And if a conjecture compared two quantities, the quantities should be close in size. Using these standards, he was able to eliminate all but 20 to 50 of the proposals produced by a run of the program.

Dr. Fajtlowicz said that at least 20 mathematicians have worked on proving conjectures generated by the program and that he knows of five papers proving Graffiti conjectures that have either been published or accepted for publication by mathematics journals.

Fan Chung, who directs research in mathematics and communications at Bell Communications Research in Morristown, N.J., has worked on a Graffiti conjecture. She described it as a statement about two properties of a graph. One is the average distance between pairs of points. The other is a quantity called the independence number, which has to do with how many areas of the graph are not adjacent to one another. The conjecture said that if the average distance is large, so is the independence number.

What attracted me to the conjecture was its simplicity, Dr. Chung said. It was a very clean relationship. But it was harder to prove than it first looked.

Dr. Chung said that given a choice, she would rather work on her own conjectures or on those that have gained a certain fame or notoriety in mathematics because they have been so difficult to verify. But, she added, It doesn’t bother me